Motivation/backrgound:
I am reading about some results in topological dynamics, and particularly concerning the action of seperable metrisable groups G acting on compact Hausdorff spaces. I am trying to digest the paper here relating extreme amenability of topological groups to structural Ramsey Theory.
Theorem 1.5 in the paper proves that the greatest G-ambit is the inverse limits of a system of metrisable G ambits.
This is done by considering the space of right uniformly continuous functions, $RUC^b(G)$, whose maximal ideal $S(G)$ is a G-ambit.
By right uniformly continuous, we mean maps $x:G\rightarrow \mathbb{C}$ such that for any $\epsilon > 0$, $\exists$ an open neighbourhood $V$ of the identity $1_G\in G$ such that $\forall g,h,\in G$ such that $gh^{-1}\in V, |x(g)-x(h)|<\epsilon$.
Since G is metrisable, and we can equip it with a right invariant compatible metric, we can also write this
$\forall \epsilon>0\exists \delta>0 $ such that $d(g,h)<\delta \implies |x(g)-x(h)|<\epsilon$
Now in this part of the proof I am struggling with on page 13, we construct our inverse system using the directed set $<I, \leq >$ where $I $ consists of all separable, closed, unital C* subalgebras of $RUC^b(G)$. We take $D\subset G$ as the countable dense subset in G (since G is separable)
Now the paper claims:
Since a closed, unital C* algebra of $RUC^b(G)$ is G-invariant iff it is D invariant.
Now that G inavriance $\implies$ D invariance is trivial. But I am not sure about proving the converse. I think it must have to do with the defintion of $RUC^b(G)$ using open sets.
Attempt:
Suppose that S is a D invariant C* subalgebra, but it is not G invariant. So there is a $g\in G-D$ such that $gS \not\subset S$
i.e. we have an $x\in S$ such that $g \bullet x \not\in S$. Note that the action of G on $RUC^b(G)$ is defined such that $g \bullet x (h) = x(g^{-1}h)$
Maybe we can then show x is not right unifromly continuous and so have our contradiction?